A non-measurable set with density $\frac{1}{2}$

Question

According to Lebesgue's Density Theorem:

Let $\mu$ be the Lebesgue outer measure, and let $A\subseteq\mathbb{R}$ be a Lebesgue measurable set. Then the limit: $$\lim_{h\to0}\frac{\mu\left(A\cap\left(x-h,x+h\right)\right)}{2h}$$ exists and equals to either 0 or 1, for almost every $x\in\mathbb{R}$.

Is there a known example of a set $A\subseteq\mathbb{R}$ for which the claim above doesn't hold?

If not, Is there a known proof of existance of such a set?

(A set like this, if exists, is clearly non-measurable by the theorem above).

Hmm, good question. I was initially going to dismiss this question until I saw that $\mu$ referred to the Lebesque outer measure instead of the standard Lebesque measure. It still might be a trivial result for some, but not me. — QC_QAOA, Jan 30 '23 at 13:21
Maybe a lead? There is that theorem from Sierpiński where he shows the existence of a non-measurable subset of $[0,1]$ the following way. Let $\mathscr{U}$ be some non-principal ultrafilter over $\mathbb{N}$; then define: $$E=\left{\sum_{n\in A}!2^{-n-1}\ \Bigg|\ A\in\mathscr{U} \right}.$$ He shows that $E$ and $F=[0,1]\smallsetminus E$ are symmetric around $\frac12$. If they were measurable they would have the same measure $\frac12$. Using a reductio ad absurdum reasoning he proves that if $E$ is measurable, then its measure must be 1. (Now, this set $E$ has an outer measure…) — jp boucheron, Jan 30 '23 at 13:39
Somewhat relevant: A converse of Lebesgue's density theorem by Shunji Kametani (1940) AND On Lebesgue's density theorem by Casper Goffman (1950) AND The construction of a Lebesgue measurable set with every density by Allan deCamp (1990-1991) AND Exceptional points for Lebesgue's density theorem on the real line by András Szenes (2011). — Dave L. Renfro, Jan 30 '23 at 15:35
About as extreme as one can get, at least for the real line, is probably a 1917 result of Luzin/Sierpinski discussed in this mathoverflow answer. Also, this MSE answer discusses Goffman's 1950 paper a little and this mathoverflow question and answer provides a follow-up to that MSE answer. — Dave L. Renfro, Jan 30 '23 at 15:47

A non-measurable set with density $\frac{1}{2}$

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